ISSN 1518-3548
Working Paper Series
Optimal Monetary Rules: the Case of Brazil
Charles Lima de Almeida, Marco Aurélio Peres, Geraldo da Silva e Souza
and Benjamin Miranda Tabak
February, 2003
ISSN 1518-3548
CGC 00.038.166/0001-05
Working Paper Series
Brasília
n. 63
Feb
2003
p. 1 – 15
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Optimal Monetary Rules: The Case of Brazil
Charles Lima de Almeida
Marco Aurélio Peres
Geraldo da Silva e Souza
Benjamin Miranda Tabak*
Abstract
Within a dynamic programming approach we derive an optimal rule for the
central bank to attain it's inflation targeting goals. The short-run nominal
interest rate is used as an instrument to achieve monetary objectives. The
model is tested for the Brazilian economy and compared with results found for
other countries. Evidence for the estimated feedback interest rule for the
Central Bank suggests that the cost of reducing inflation in an open economy is
lower than that of a closed economy.
JEL Classification: E43, E52.
Keywords: optimal Taylor rule, monetary policy, inflation targeting.
Resumo
Através de técnicas de programação dinâmica derivamos uma regra ótima para
o Banco Central atingir suas metas de inflação. A taxa nominal de juros é
utilizada como instrumento para atingir os objetivos de política monetária. O
modelo é testado para a economia brasileira e comparam-se os resultados com
encontrados para outros países. Evidência para regra de feedback encontrada
sugere que os custos de reduzir a inflação em economias abertas é menor do
que em economias fechadas.
*
Research Department, Central Bank of Brazil. Corresponding author’s e-mail address:
[email protected]
3
1.
Introduction
Recently, several countries have been adopting a target inflation framework for monetary
policy. New Zealand, Canada and United Kingdom have decided to employ an inflation
targeting framework in the conduit of monetary policy and have successfully reduced their
inflatio n rates and gained control on inflation1 . It is often argued that an independent
central bank is a key element for a successful monetary policy. The good performance in
the maintenance of low inflation rates that the National Bank of Switzerland and the
Bundesbank have had is mainly attributed to their high level of independence, and it has
certainly strongly influenced the position of the system of European central banks toward
an independent central bank.
Following this trend, other countries (e.g. Chile, Mexico, Argentina, Spain and France)
have been granting a greater independence for their central banks allowing them to
conduce monetary policy with much less government interference. The main goal is price
stability and a common point among these countries has been the adoption of an inflation
targeting framework. In Brazil, since mid-1999, six months after abandoning the fixed
exchange rate regime, the Central Bank of Brazil has adopted an inflation targeting regime
for the conduit of monetary policy.
The main purpose of this paper is to derive an optimal feedback rule based on the model
proposed by Ball (1998) and to estimate a short run reaction function for interest rates for
Brazil, for an open economy. The main assumption is that interest rates are the central
bank's main instrument to reduce inflation and the level of activity.
Normally, models estimating IS-AS-type models use OLS regressions. In this paper we
derive an optimal feedback rule and use the estimated coefficients from IS-AS equatio ns to
find an empirical relation between central bank's instrument and macroeconomic variables
such as inflation, output gap and exchange rates. We suggest the use of two stage least
squares using adequate instrumental variables to estimate these equations to overcome
problems inherent to the nature of Brazilian macroeconomic variables 2 . We also compare
our results with those found in the literature.
The plan of the paper is as follows. In the first section we derive a Taylor rule for the
Brazilian economy. Section two presents empirical results. In the last section we conclude
and give directions for further research.
2.
The model
In this section an optimal monetary rule is derived for the Brazilian economy. We use the
following IS equation:
yt+1 = a 1 yt − a2 it + a3et + ut +1
(2.1)
where yt stands for the output gap, it is the real interest rate, et is the real exchange rate,
1
For an interesting analysis of inflation targeting see Walsh (2001).
Using instrumental variables for the terms containing lagged inflation is crucial
because these terms are correlated with the residuals.
2
4
and ut is a demand shock, assumed to be normally distributed.
The supply curve is represented by the traditional Phillips curve:
π t +1 = πt + γ yt + µ (ε t − ε t−1 ) + ηt +1
(2.2)
where π t is the inflation rate, ∆ε t is the depreciation rate in the nominal exchange rate
and ηt +1 is the supply shock not correlated with u t +1 .
The policy maker chooses in instant t the interest rate it , and the state variable in instant t
is:
zt = γ yt + π t + µ ( ε t − ε t−1 )
(2.3)
The optimal feedback rule will be given by:
θ t = Xz t
(2.4)
where
θ t = a1 ytt + a2it + µ et
(2.5)
Equations (1) and (2) can be rewritten as:
y t+1 = θ t + ut +1
(2.6)
and
π t+1 = z t + ηt +1
(2.7)
We assume that the central bank's loss function is given by:
L=
∞
1
i
2
2
Et ∑ β  λ yt +1 + π t+1 
2 i =1
(2.8)
The objective of the policy maker is to minimize this loss function subject to:
z t +1 = z t + γθ t + ηt +1 + γut +1
(2.9)
Define the value function as:
1

2
2
V ( z t ) = min Et  ( λy t+1 + π t +1 ) + βV ( z t+1 ) 
2

(2.10)
replacing (2.6), (2.7) and (2.9) in the value function we obtain:
5
1
1

2
2
 λ Et (θ t + ut+1 ) + Et ( zt + ηt +1) + 
V ( zt ) = min  2
2

θt
 β EtV ( z t + γθ t + ut +1 + γηt +1)

(2.11)
Solving problem (2.11) with respect to θ t gives the first order condition:
λθ t + γβVz Et ( z t +1 ) = 0
(2.12)
Applying the envelope theorem with respect to z t obtains:
Vz ( zt ) = zt + β Vz Et (z t+1 )
(2.13)
Multiplying (2.13) by γ , substituting in (2.12), taking this expression one-step forward and
the expectations
EtV z ( zt +1 ) = z t + γθ t −
λ
Et (θ t+1 )
γ
(2.14)
Replacing (2.14) in (2.12):
θt = −
γβ
βλ
zt +
Et (θ t+1 )
2
λ +γ β
λ + βγ 2
(2.15)
When the policy is established in instant t, z t is the state variable and thus the optimal
policy rule has a quadratic form θ t = Xz t . Therefore
E t+1 (θ t +1 ) = XE t ( z t +1 ) = X (1 + γX )z t
(2.16)
replacing this expression in (2.15) obtains the following quadratic form:
(
)
λβγX 2 − λ − βλ + γ 2 β X + γβ = 0
(2.17)
Stability requires 1 + 2γ (1 + α ) X < 1 . Hence, the solution for (2.17) is given by:
X
=
( λ − βλ + γ 2 β ) ±
( λ − βλ + γ 2 β ) 2 + 4 (γ 2 β 2 λ )
2 βγλ
(2.18)
remembering that:
z t +1 = z t + γθ t = ( Xγ + 1) zt
(2.19)
After some algebraic operations the product of the roots is:
6
X1X 2 = −
1
<0
λ
(2.20)
The root of interest is the one that satisfies the stability condition, that is the negative root
X2 . Finally, replacing X2 in (2.4) gives
( λ − βλ + γ 2 β ) − (λ − βλ + γ 2 β ) 2 + 4(γ 2 β 2 λ )
θt =
zt
2 βγλ
(2.21)
We can derive the optimal rule for the interest rate
it =
3.
a1−γ X 2
a
X
X
yt + 3 ∆et + 2 π t + η 2 et
a2
a2
a2
a2
(2. 22)
Empirical Results
For the econometric analysis we have used quarterly data and our sample begins in the first
quarter of 1994 and ends in the last quarter of 2001. All variables are in natural logs. As a
proxy for the output gap we have estimated a Hodrick-Prescott filter and used the
difference between observed GDP and the filtered series. The inflation rate is given by
IPCA. The interest rate is given by SELIC which is the instrument that the central bank
uses to achieve it's price stability goals.
According to the results found in table 1 both the lag of the output gap and lagged interest
rate are significant in explaining current output gap, and the sign of the coefficients are in
line with the expected sign. We used as instruments a dummy for the Russian crisis, three
lags for the interest rate and four lags for the government spending.
Table 1. IS equation - Closed Economy
Variables
Coefficients
0.34**
y t −1
(0.1697)
-0.06*
it
(0.0117)
Adjusted R2 = 77%
p-value
0.04
0.00
Standard errors are given in parenthesis
* Rejection of the null with 99% confidence
** Rejection of the null with 95% confidence
Table 2 presents empirical results for the Phillips equation. Both lagged output gap and
inflation are significant in explaining current inflation. We have used as instruments a
dummy for the Russian crisis, six lags for the inflation rate and two lags for government
spending.
7
Table 2. Phillips equation - Closed Economy
Variables
Coefficients
0.34**
y t −1
(0.014)
0.60*
π t−1
(0.023)
Adjusted R2 = 90%
p-value
0.029
0.000
Standard errors are given in parenthesis
* Rejection of the null with 99% confidence
** Rejection of the null with 95% confidence
Applying Augmented Dickey and Fuller tests the null of a unit root for the output gap,
interest rates and inflation is rejected. Results for these unit roots are available upon
request from the authors.
In order to derive the optimal policy rule for the Brazilian economy we assumed β =0.7
and λ =1, which are the intertemporal discount factor and the relative weight of output gap
in the loss function. After replacing this parameters and coefficients in table 1 and 2 one
obtains
it = 5,5 yt + 4,2π t
(3.1)
Our results are quite different from those found in Taylor (1993) and Ball (1998) and are
more in line with those found in Walsh (1997).
Table 3. Comparison of optimal rules - Closed Economy
Coefficients
yt
πt
Taylor (1993)
0.50
1.50
Walsh (1997)
4.37
1.26
Authors
5.5
4.2
Ball (1998)
0.80
1.46
yt−1
1.59
-
The coefficient on the output gap is similar to that found in Ball (1998), while the
coefficient on inflation is much higher than that of the rest, which suggests that the central
bank of Brazil has to increase it's interest rates much more than developed countries in
order to counterbalance an increase in inflation.
In tables 4 and 5 we present results for an open economy.
8
Table 4. IS equation - Open Economy
Variables
Coefficients
0.36**
y t −1
(0.168)
-0.04**
e t −2
(0.018)
0.06*
it
(0.014)
Adjusted R2 = 75%
p-value
0.04
0.05
0.00
Standard errors are given in parenthesis
* Rejection of the null with 99% confidence
** Rejection of the null with 95% confidence
Table 5. Phillips equation - Open Economy
Variables
Coefficients
0.08**
y t −1
(0.05)
0.07**
∆ε t
(0.01)
0.65*
π t−1
(0.038)
2
Adjusted R = 64%
p-value
0.02
0.03
0.00
Standard errors are given in parenthesis
* Rejection of the null with 99% confidence
** Rejection of the null with 95% confidence
Replacing the results one obtains the optimal rule:
i t = 5.2 y t + 0.3π t + 0.6ε t −1 + 0.2∆et
(3.2)
To the best of our knowledge, most research on developed countries has estimated
different optimal feedback rules, making comparisons more difficult. As we can see, the
coefficient on inflation has decreased to 0.3. Thus, the nominal interest rate is increased
more than five-to-one with increases in output gap. The cost to reduce inflation seems to be
lower in an open economy, which is an argument in favor of commercial liberalization.
4.
Conclusions
In this paper we have presented an optimal policy rule for the central bank to achieve it's
monetary policy goals, derived using a dynamic programming approach and a dynamic
loss function. We estimated IS-AS equations using two stage least squares and fitted an
optimal feedback rule for short run interest rates for both a closed and open economy.
We have found that the feedback rule behaves differently from similar rules estimated for
developed countries. For the open economy evidence suggests that interest rates needs to
raise less than a one-to-one with inflation (while the contrary happens within a closed
economy). Thus, it is found that within an open economy the central banks have much
more power to reduce inflation than within a closed economy. This issue will be left for
further research.
9
References
Ball, L. (1998) Policy Rules for open Economies, NBER Working Paper 5952, October.
Taylor, J. B. (1993) Discretion versus Policy Rules in Practice, Carnegie-Rochester
Conferences on Public Policy, 39, December, 95-214.
Walsh, C. E. (1997) Monetary Theory and Police, MIT Press: Chicago.
Walsh, C.E. (2001) Teaching Inflation Targeting: An analysis for intermediate macro,
University of California. Forthcoming in Journal of Economic Education.
10
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